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Geometric Control of a Quadrotor UAV Transporting a Payload Connected via Flexible Cable
Farhad A. Goodarzi, Daewon Lee, Taeyoung Lee
TL;DR
Existing load-transport models often assume a taut, rigid cable and neglect cable and payload dynamics, limiting their applicability when a cable deforms. This paper derives coordinate-free dynamics for a quadrotor carrying a serial-link cable load and designs coupled geometric control; its simulations and experiments report stabilization and reduced link oscillation.
Problem
Many existing approaches assume a taut, rigid cable and neglect cable and payload dynamics, limiting their applicability to aggressive transport with cable deformation or low tension.
Source: §1 · p. 1Method
The paper derives coordinate-free Lagrangian dynamics for an arbitrary serial-link cable and develops geometric nonlinear control that explicitly accounts for cable–load coupling.
Source: §1 · pp. 1–2Results
Takeaways & Limitations
The equations are globally defined for arbitrary link count and avoid local-parameterization singularities, supporting coordinate-free control design for agile maneuvers.
Source: §1 · pp. 1–2
Abstract
from arXiv · showhide
We derived a coordinate-free form of equations of motion for a complete model of a quadrotor UAV with a payload which is connected via a flexible cable according to Lagrangian mechanics on a manifold. The flexible cable is modeled as a system of serially-connected links and has been considered in the full dynamic model. A geometric nonlinear control system is presented to exponentially stabilize the position of the quadrotor while aligning the links to the vertical direction below the quadrotor. Numerical simulation and experimental results are presented and a rigorous stability analysis is provided to confirm the accuracy of our derivations. These results will be particularly useful for aggressive load transportation that involves large deformation of the cable.
1. INTRODUCTION
Existing load-transport models commonly assume a taut, rigid cable and neglect cable and payload dynamics, limiting their stated applicability to aggressive transport with cable deformation. This paper models serial cable links and develops coordinate-free dynamics and coupled nonlinear control for the quadrotor and load.
1. INTRODUCTION
- Rigid, taut-cable assumptions and neglected cable and payload dynamics limit prior approaches when aggressive transport deforms the cable or lowers its tension.The paper notes that these approaches also make safety operations impossible to guarantee in those conditions.Source: §1 · p. 1
- The cable is modeled as an arbitrary number of serial links, yielding equations on a configuration manifold formed from quadrotor pose and one two-sphere per link.The links may have different sizes and masses and are connected by spherical joints.Source: §1 · p. 1
- The proposed geometric controller accounts explicitly for cable–load and quadrotor coupling while stabilizing the hanging links and translating the quadrotor to a desired position.The links' force and moment effects are not treated merely as disturbances.Source: §1 · p. 2
- Coordinate-free dynamics and controllers yield compact expressions, avoid local-parameterization singularities, and support uniform agile maneuvers.The paper contrasts these expressions with local-coordinate formulations that can require symbolic computational tools for complex multibody systems.Source: §1 · p. 2
- The paper adds Lyapunov stability analysis without timescale-separation assumptions and an integral control term for robustness to unstructured rotational and translational uncertainties.These are presented in comparison with the authors' preliminary results in.Source: §1 · p. 2
A FLEXIBLE CABLE
The flexible cable is represented by serial links whose directions lie on two-spheres, and the model's Lagrangian yields globally defined, coordinate-free equations of motion for any link count.
A FLEXIBLE CABLE
- Each link is represented by a unit direction from the quadrotor toward the payload, with its mass concentrated at the link's outboard end.The resulting configuration manifold is SO(3) × R3 × (S2)^n.Source: A FLEXIBLE CABLE · p. 2
- The Lagrangian is formed from the quadrotor and link kinetic energies and the gravitational potential energy.The paper derives the Lagrangian as total kinetic energy minus potential energy.Source: §2.1 · pp. 2–3
- The Euler–Lagrange equations are derived on R3 × SO(3) × (S2)^n and are globally defined, compact, and uniform for any number of links.The coordinate-free form avoids local-coordinate singularities and is presented as suitable for control design.Source: §2.2 · p. 3
3. CONTROL SYSTEM DESIGN FOR A SIMPLIFIED DYNAMIC MODEL
This section formulates stabilization of the hanging-link equilibrium at a desired quadrotor position using a simplified model. It linearizes about that equilibrium and chooses feedback gains to stabilize the linearized and then nonlinear dynamics.
3. CONTROL SYSTEM DESIGN FOR A SIMPLIFIED DYNAMIC MODEL
- The control objective is to stabilize the hanging-link equilibrium while placing the quadrotor at a fixed desired location.At equilibrium, all links point downward.Source: §3.1 · p. 4
- The simplified model replaces the thrust force with a fictitious three-dimensional input, assumes instantaneous attitude control, and ignores Δx.The paper introduces this simplification before incorporating attitude dynamics in the next section.Source: §3.2 · p. 4
- The fictitious input is designed from dynamics linearized about the hanging equilibrium, using link-direction and angular-velocity variations in the tangent plane.The third components of the link variation and angular-velocity variation are zero at equilibrium and are omitted from the linearized state.Source: §3.3 · p. 4
- If the linearized system is controllable, gains can be chosen to make its equilibrium asymptotically stable, which yields asymptotic stability for the nonlinear equations.The section also states that the gains may be chosen so the closed-loop matrix A is Hurwitz.Source: §3.3 · p. 4
4. CONTROLLER DESIGN FOR A QUADROTOR WITH A FLEXIBLE CABLE
The controller steers the quadrotor’s total thrust toward a fictitious ideal force while controlling its attitude, and explicitly accounts for cable–vehicle coupling. The paper establishes tracking stability with Lyapunov analysis and adds integral terms for disturbance robustness.
Controller design
- Attitude control makes the thrust direction follow the fictitious input, while suitable thrust magnitude drives total thrust toward the ideal force for full-model stabilization.The ideal total thrust is defined for stabilizing the desired equilibrium.Source: §4 · p. 5, §4.1 · p. 5
- A saturated translational integral term is added to eliminate full-model disturbance effects, with a known upper bound assumed for uncertainty.The saturation is applied elementwise to vector inputs.Source: §4.1 · p. 5
- The desired third body-axis direction constrains attitude, while a projected desired first-axis direction resolves the remaining yaw freedom.The construction assumes the desired first axis is not parallel to the commanded third axis.Source: §4.1 · p. 5
- An integral term is also introduced to eliminate fixed rotational disturbance, and the attitude controller uses positive gains.The passage identifies the gains as kR, kΩ, and kI.Source: §4.1 · p. 5
Stability analysis
- Lyapunov analysis establishes stable tracking, asymptotic convergence of attitude and translational tracking errors, and bounded integral terms.The paper states that this analysis requires no timescale separation assumptions or singular perturbation, and describes the integral term as providing robustness to unstructured rotational and translational uncertainty.Source: §4.2 · p. 5
5. NUMERICAL EXAMPLE
The numerical example compares the controller with and without its integral term under fixed disturbances. The integral-term case significantly reduces steady-state error while stabilizing the payload and links beneath the quadrotor.
Simulation setup
- The simulation uses a 0.5 kg quadrotor and five identical links, each with mass 0.1 kg and length 0.1 m.The quadrotor inertia is J = diag[0.557, 0.557, 1.05]×10−2 kgm2.Source: §5 · p. 6
- The links begin curved along the horizontal direction, with each link’s initial angular velocity set to zero.Source: §5 · p. 6
Evaluation measures
- The stated link-stabilization errors measure direction error and angular velocity error.The simulation reports reductions in both errors.Source: §5 · p. 6
Simulation results
- The integral terms significantly eliminate steady-state error under fixed disturbances as the quadrotor position converges to its desired value while stabilizing the payload and links.The passage compares cases with and without the integral term.Source: §5 · p. 6
6. EXPERIMENTAL RESULTS
Experiments compare a position-only quadrotor controller with the proposed controller on a quadrotor carrying a cable-suspended payload. The proposed controller reduces unwanted link oscillation in the reported comparison.
6. EXPERIMENTAL RESULTS
- The test vehicle weighed 0.791 kg and carried a 0.036 kg payload on a 0.7 m cable.Vehicle position was measured with Vicon, and velocity was estimated from that measurement.Source: §6 · pp. 6–7
- The onboard controller ran at 120 Hz, while the Vicon thread estimated vehicle velocity at 30 Hz.The IMU thread received inertial measurements and estimated angular velocity.Source: §6 · p. 7
- The experiment compared the proposed controller with a position controller that did not include payload or link dynamics.The baseline controller was used to hover at the desired location.Source: §6 · p. 7
- The proposed controller reduced undesired link oscillation relative to quadrotor position control that omitted payload and link dynamics.Figures 6 and 7 show the position and link-deflection comparisons.Source: §6 · p. 7, Figure 6 · p. 8, Figure 7 · p. 8
7. CONCLUSIONS
The conclusion presents a 3D Euler–Lagrange model of a quadrotor carrying a flexible cable modeled as a chain pendulum. It pairs the model with a geometric nonlinear controller that stabilizes the links below the quadrotor without local angle coordinates.
7. CONCLUSIONS
- Euler–Lagrange equations model a quadrotor transporting a load on a flexible cable in 3D.The derivation accommodates an arbitrary number and configuration of links.Source: §7 · p. 8
- A geometric nonlinear controller stabilizes the links below the quadrotor from any chosen initial condition.The target is the equilibrium position.Source: §7 · p. 8
- The derivation avoids local angle coordinates and is presented as a compact form for arbitrary link configurations.The conclusion describes this coordinate-free approach as advantageous.Source: §7 · p. 8
APPENDIX A
Appendix A develops the equations and stability proof using Lagrangian variations and Lyapunov analysis. It combines attitude and translational-link error analyses to establish convergence of the tracking errors under the stated gain condition.
APPENDIX A
- The appendix derives equations of motion from the Lagrangian using the Lagrange–d’Alembert principle and variations.It then rewrites the link equations in terms of angular velocities.Source: Appendix A · pp. 8–9
- The stability proof treats rotational dynamics first, then combines them with translational quadrotor and link dynamics.The translational analysis derives tracking-error dynamics and a Lyapunov function for the quadrotor and links.Source: Appendix A · p. 9, c) Translational Error Dynamics · p. 10
- With γ < λmin(W)/(2∥P∥²), the tracking-error equilibrium is Lyapunov stable, the error and integral-control variables remain bounded, and z converges to zero.The proof uses the LaSalle–Yoshizawa theorem for convergence.Source: e) Lyapunov Candidate for the Complete… · p. 12
- The derivation approximates ξ by e3 × q when developing the higher-order terms.Figure 8 illustrates the relevant directions.Source: §4 · p. 12, Figure 8 · p. 12